Intuition for COM

Q. Three particles of the same mass lie in the x–y plane. The (x, y) coordinates of their positions are (1, 1), (2, 2) and (3, 3) respectively. The (x, y) coordinates of the centre of mass are

1. (1, 2)
2. (2, 2)
3. (4, 2)
4. (6, 6)

Q. Three identical metal balls each of radius $r$ are placed touching each other on a horizontal surface such that an equilateral triangle is formed. When the centre of the three balls are joined, centre of mass of the system is located at

1. horizontal surface.
2. centre of one of the balls.
3. line joining centre of any two balls.
4. point of intersection of the medians of the triangle

Q. In the CO molecule, the centre of mass is at a distance of 0.068 nm from the carbon atom. The separation between the nuclei of carbon and oxygen is

1. $\frac{28\times 0.068}{16}nm$
   
2. $\frac{28\times 0.068}{12}nm$
3. $\frac{16\times 0.068}{28}nm$
   
4. $\frac{12\times 0.068}{28}nm$

Q. A system consists of masses $M$ and $m$ (<< $M$). The centre of mass of the system is

1. at the middle.
2. nearer to $M$.
3. nearer to $m$.
4. at the position of larger mass.

Q. Three particles of masses $50\text{g},100\text{g}$ and $150\text{g}$ are placed at the vertices of an equilateral triangle of side $2\text{m}$ as shown in the figure. The $(x,y)$ co-ordinates of the centre of mass (in $\text{m})$ will be

1. $(\frac{7}{6},\frac{\sqrt{3}}{2})$
2. $(\frac{7}{12},\frac{\sqrt{3}}{4})$
3. $(\frac{7}{3},\sqrt{3})$
4. $(\frac{7}{12},\frac{\sqrt{3}}{2})$

Q.

lf all the particles of a system lie in X-Y plane, is it necessary that the centre of mass be in X-Y plane?

1. Yes

2. Never

3. Some times

4. Not theoretically but practically

Q. Figure shows a uniform disc of radius $R$, from which a hole of radius $R/2$ has been cut out from left of the centre and is placed on the right of the centre of the disc. Find the COM of the resulting disc.

1. $(\frac{R}{2},0)$
   
2. $(\frac{R}{4},0)$
3. $(\frac{R}{2},\frac{R}{2})$
4. $(\frac{R}{4},\frac{R}{4})$

Q.

Four particles $A$, $B$, $C$ and $D$ having masses $m$, $2m$, $3m$ and $4m$ respectively are placed in order at the corners of a square of side $a$. Locate the centre of mass

1. $\frac{a}{2}$ , $\frac{a}{2}$

2. $\frac{a}{2}$, $\frac{7a}{10}$

3. $\frac{a}{2}$, $\frac{3}{4}$

4. None of these

Q. A proton is fired from origin with velocity $\overrightarrow{v}=v_0\hat{j}+v_0\hat{k}$ in a uniform magnetic field $\overrightarrow{B}=B_0\hat{j}$ (Take $m=$mass and $q=$ charge). Path of the proton particle is

1. A helix with non-uniform pitch
2. A circle
3. A straight line
4. A helix with uniform pitch